Questions: Compute each sum below. If applicable, write your answer as a fraction. (3)+(3)^2+(3)^3+...+(3)^7=9840 ∑(i=1)^(8) 3(-1/2)^i=-1

Solution Steps

To solve the given sums, we need to recognize the type of series they represent and use the appropriate formulas.

1. The first sum is a geometric series with the first term \(a = 3\) and common ratio \(r = 3\). The sum of the first \(n\) terms of a geometric series can be calculated using the formula: \[ S_n = a \frac{r^n - 1}{r - 1} \]

2. The second sum is also a geometric series with the first term \(a = 3 \left(\frac{-1}{2}\right)\) and common ratio \(r = \frac{-1}{2}\). The sum of the first \(n\) terms of a geometric series can be calculated using the same formula: \[ S_n = a \frac{r^n - 1}{r - 1} \]

The first sum is a geometric series with the first term \(a = 3\) and common ratio \(r = 3\). The second sum is also a geometric series with the first term \(a = 3 \left(\frac{-1}{2}\right)\) and common ratio \(r = \frac{-1}{2}\).

For a geometric series, the sum of the first \(n\) terms is given by: \[ S_n = a \frac{r^n - 1}{r - 1} \]

For the first sum: \[ a_1 = 3, \quad r_1 = 3, \quad n_1 = 7 \] \[ S_1 = 3 \frac{3^7 - 1}{3 - 1} \] \[ S_1 = 3 \frac{2187 - 1}{2} \] \[ S_1 = 3 \frac{2186}{2} \] \[ S_1 = 3 \times 1093 \] \[ S_1 = 3279 \]

For the second sum: \[ a_2 = 3 \left(\frac{-1}{2}\right) = -1.5, \quad r_2 = \frac{-1}{2}, \quad n_2 = 8 \] \[ S_2 = -1.5 \frac{\left(\frac{-1}{2}\right)^8 - 1}{\frac{-1}{2} - 1} \] \[ S_2 = -1.5 \frac{\frac{1}{256} - 1}{-\frac{3}{2}} \] \[ S_2 = -1.5 \frac{\frac{1 - 256}{256}}{-\frac{3}{2}} \] \[ S_2 = -1.5 \frac{\frac{-255}{256}}{-\frac{3}{2}} \] \[ S_2 = -1.5 \times \frac{-255}{256} \times \frac{2}{3} \] \[ S_2 = -1.5 \times \frac{-255 \times 2}{256 \times 3} \] \[ S_2 = -1.5 \times \frac{-510}{768} \] \[ S_2 = -1.5 \times -0.6641 \] \[ S_2 = 0.9961 \]

Final Answer